Do scientists have to record all data precisely in order to follow the scientific method?
The scientific method should allow other scientists to duplicate the work that has been done, so that they can check the results and observations. New science is accepted by the community when it is demonstrated to be repeatable and reproducible.
This means that the method must be recorded with sufficient clarity for other to reproduce the experiment, and any measurements must be recorded sufficiently precisely for others to check whether their own results are 'the same'.
This means scientists have to understand the sources if error in their experiment and seek to minimise these or explore the range of independent variables for which their experiment is valid. Once the sources of error are known they can be quantified or estimated, and measurements can then be made and recorded with a precision which is appropriate to the overall precision of the experiment.
For example, there is no point in weighing out the mass of magnesium you are going to react with excess hydrochloric acid to four decimal places if the thermometer you are reading the temperature change with only reads to the nearest degree.